Recovering a Holographic Geometry from Entanglement Sebastian - - PowerPoint PPT Presentation

Context Holographic EE Geometric Bulk Reconstruction Extensions

Recovering a Holographic Geometry from Entanglement

Sebastian Fischetti

1904.04834 with N. Bao, C. Cao, C. Keeler 1904.08423 with N. Engelhardt

• ngoing with N. Bao, C. Cao, J. Pollack, P. Sabella-Garnier

McGill University

UT Austin October 29, 2019

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Quantum Gravity from AdS/CFT

An ambitious question The (semi)classical gravity we observe in our universe emerges from some more fundamental quantum theory - how?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Quantum Gravity from AdS/CFT

An ambitious question The (semi)classical gravity we observe in our universe emerges from some more fundamental quantum theory - how? Hard to even begin to answer because we don’t know what the full formulation of such a theory is! We need a framework in which to work: in context of string theory, AdS/CFT gives us a nonperturbative, indirect definition

• f a theory of quantum gravity

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Quantum Gravity from AdS/CFT

AdS/CFT Correspondence [Maldacena] A nonperturbative, background-independent theory of quantum gravity with asymptotically (locally) anti-de Sitter boundary conditions – the “bulk” – is dual to a conformal field theory – the “boundary” – living on (a representative of the conformal structure

• f) the asymptotic boundary of the bulk.

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Quantum Gravity from AdS/CFT

AdS/CFT Correspondence [Maldacena] A nonperturbative, background-independent theory of quantum gravity with asymptotically (locally) anti-de Sitter boundary conditions – the “bulk” – is dual to a conformal field theory – the “boundary” – living on (a representative of the conformal structure

• f) the asymptotic boundary of the bulk.

Work around a limit in which the bulk is well-approximated by a classical geometry: ← → AdS CFT

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

The Holographic Dictionary

Using AdS/CFT as a framework, we can refine the question: A slightly less vague question In AdS/CFT, when and how does (semi)classical gravity emerge from the boundary field theory?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

The Holographic Dictionary

Using AdS/CFT as a framework, we can refine the question: A slightly less vague question In AdS/CFT, when and how does (semi)classical gravity emerge from the boundary field theory? Requires understanding what “dual” means: the holographic dictionary

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

The Holographic Dictionary

Using AdS/CFT as a framework, we can refine the question: A slightly less vague question In AdS/CFT, when and how does (semi)classical gravity emerge from the boundary field theory? Requires understanding what “dual” means: the holographic dictionary Going from the bulk to the boundary is pretty well-understood (e.g. one-point functions of local boundary operators are given by the asymptotic behavior of local bulk fields) Going from the boundary to the bulk is harder: this is broadly termed “bulk reconstruction”

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Line of Attack

The (semi)classical gravity we observe in our universe emerges from some more fundamental quantum theory - how? ⇓ (AdS/CFT) In AdS/CFT, how do the CFT degrees of freedom rearrange themselves to look like a gravitational theory?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Line of Attack

The (semi)classical gravity we observe in our universe emerges from some more fundamental quantum theory - how? ⇓ (AdS/CFT) In AdS/CFT, how do the CFT degrees of freedom rearrange themselves to look like a gravitational theory? ⇓ (classical limit) When and how does (semi)classical gravity emerge from the boundary field theory?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Line of Attack

The (semi)classical gravity we observe in our universe emerges from some more fundamental quantum theory - how? ⇓ (AdS/CFT) In AdS/CFT, how do the CFT degrees of freedom rearrange themselves to look like a gravitational theory? ⇓ (classical limit) When and how does (semi)classical gravity emerge from the boundary field theory? ⇓ (probe limit) How are operators on a fixed bulk geometry recovered?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Line of Attack

The (semi)classical gravity we observe in our universe emerges from some more fundamental quantum theory - how? ⇓ (AdS/CFT) In AdS/CFT, how do the CFT degrees of freedom rearrange themselves to look like a gravitational theory? ⇓ (classical limit) When and how does (semi)classical gravity emerge from the boundary field theory? ⇓ (probe limit) How are operators on a fixed bulk geometry recovered?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Reconstruction of Bulk Operators

In pure AdS, local field operators can be expressed in terms of local boundary

• perators by integrating against a kernel

[Hamilton, Kabat, Lifschytz, Lowe]:

φ(X) =

• D⊂∂M

dd−1x K(X|x)O(x)

b

D X

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Reconstruction of Bulk Operators

In pure AdS, local field operators can be expressed in terms of local boundary

• perators by integrating against a kernel

[Hamilton, Kabat, Lifschytz, Lowe]:

φ(X) =

• D⊂∂M

dd−1x K(X|x)O(x) Kernel may be taken to have support on different boundary regions D

b

D X

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Reconstruction of Bulk Operators

In pure AdS, local field operators can be expressed in terms of local boundary

• perators by integrating against a kernel

[Hamilton, Kabat, Lifschytz, Lowe]:

φ(X) =

• D⊂∂M

dd−1x K(X|x)O(x) Kernel may be taken to have support on different boundary regions D

b

D X WRindler[D]

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Reconstruction of Bulk Operators

In pure AdS, local field operators can be expressed in terms of local boundary

• perators by integrating against a kernel

[Hamilton, Kabat, Lifschytz, Lowe]:

φ(X) =

• D⊂∂M

dd−1x K(X|x)O(x) Kernel may be taken to have support on different boundary regions D Hints at subregion/subregion duality: a given boundary diamond D can reconstruct

• perators in some subregion of the bulk

b

D X WRindler[D]

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Reconstruction of Bulk Operators

In pure AdS, local field operators can be expressed in terms of local boundary

• perators by integrating against a kernel

[Hamilton, Kabat, Lifschytz, Lowe]:

φ(X) =

• D⊂∂M

dd−1x K(X|x)O(x) Kernel may be taken to have support on different boundary regions D Hints at subregion/subregion duality: a given boundary diamond D can reconstruct

• perators in some subregion of the bulk

Stronger hint comes from entanglement entropy

b

D X WRindler[D]

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Holographic Entanglement Entropy

HRT Formula [Ryu, Takayanagi, Hubeny, Rangamani] If ρR = TrR ρ is the reduced state associated to some region R and the bulk is well-approximated by a classical geometry

• beying Einstein gravity, then

S[R] ≡ − Tr(ρR ln ρR) = Area[XR] 4G , where XR is the smallest-area codimension-two extremal surface anchored to ∂R.

CFT R XR t

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Holographic Entanglement Entropy

HRT Formula [Ryu, Takayanagi, Hubeny, Rangamani] If ρR = TrR ρ is the reduced state associated to some region R and the bulk is well-approximated by a classical geometry

• beying Einstein gravity, then

S[R] ≡ − Tr(ρR ln ρR) = Area[XR] 4G , where XR is the smallest-area codimension-two extremal surface anchored to ∂R.

CFT R XR t

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Holographic Entanglement Entropy

XR is generically spacelike separated from the causal diamond D[R], so R is sensitive to more of the bulk than expected from just causal structure

CFT R XR t

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Holographic Entanglement Entropy

XR is generically spacelike separated from the causal diamond D[R], so R is sensitive to more of the bulk than expected from just causal structure Ideas from quantum error correction show that XR defines the region of the bulk to which R is sensitive: bulk operators in the entanglement wedge defined by XR can be represented by CFT operators in D[R]

[Dong, Harlow, Wall; Faulkner, Lewkowycz]

CFT R XR t

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Holographic Entanglement Entropy

XR is generically spacelike separated from the causal diamond D[R], so R is sensitive to more of the bulk than expected from just causal structure Ideas from quantum error correction show that XR defines the region of the bulk to which R is sensitive: bulk operators in the entanglement wedge defined by XR can be represented by CFT operators in D[R]

[Dong, Harlow, Wall; Faulkner, Lewkowycz]

CFT R XR t

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Holographic Entanglement Entropy

XR is generically spacelike separated from the causal diamond D[R], so R is sensitive to more of the bulk than expected from just causal structure Ideas from quantum error correction show that XR defines the region of the bulk to which R is sensitive: bulk operators in the entanglement wedge defined by XR can be represented by CFT operators in D[R]

[Dong, Harlow, Wall; Faulkner, Lewkowycz]

What about recovering the bulk geometry itself and its properties?

CFT R XR t

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Moving Up

The (semi)classical gravity we observe in our universe emerges from some more fundamental quantum theory - how? ⇓ (AdS/CFT) In AdS/CFT, how do the CFT degrees of freedom rearrange themselves to look like a gravitational theory? ⇓ (classical limit) When and how does (semi)classical gravity emerge from the boundary field theory? ⇓ (probe limit) How are operators on a fixed bulk geometry recovered?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Recovering the Geometry

The HRT formula clearly connects bulk geometry to boundary entanglement, and its key role in recovering bulk operators on a fixed background strongly suggests it should play a role in recovering the geometry as well [Van Raamsdonk]. Does it?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Recovering the Geometry

Some partial progress: Dynamics: For perturbations of vacuum, HRT implies the perturbative Einstein equations in the bulk [Lashkari, Faulkner, Guica,

Hartman, McDermott, Myers, Van Raamsdonk, ...]

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Recovering the Geometry

Some partial progress: Dynamics: For perturbations of vacuum, HRT implies the perturbative Einstein equations in the bulk [Lashkari, Faulkner, Guica,

Hartman, McDermott, Myers, Van Raamsdonk, ...]

Gravitational thermodynamics: generic area laws in the bulk can be related to monotonicity properties of entropy

A coarse-grained entropy defined by fixing a portion of the bulk geometry gives area law along (spacelike) foliations of apparent horizons [Engelhardt, Wall] Casini-Huerta c-theorem relates to mixed-signature area laws in bulk, including along early-time event horizons of black holes formed from collapse [Engelhardt, SF]

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Recovering the Geometry

Some partial progress: Dynamics: For perturbations of vacuum, HRT implies the perturbative Einstein equations in the bulk [Lashkari, Faulkner, Guica,

Hartman, McDermott, Myers, Van Raamsdonk, ...]

Gravitational thermodynamics: generic area laws in the bulk can be related to monotonicity properties of entropy

A coarse-grained entropy defined by fixing a portion of the bulk geometry gives area law along (spacelike) foliations of apparent horizons [Engelhardt, Wall] Casini-Huerta c-theorem relates to mixed-signature area laws in bulk, including along early-time event horizons of black holes formed from collapse [Engelhardt, SF]

Causal structure: instead of EE, can use the singularity structure

• f boundary correlators to deduce the causal structure of (part
• f) the causal wedge of the bulk [Engelhardt, Horowitz; Engelhardt, SF]

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Recovering the Geometry

Here, I’m interested in a more fine-grained question: Does knowledge of the entanglement entropy of all regions (i.e. the areas of all HRT surfaces) determine the bulk geometry? How? Obviously EE can’t recover the full geometry, since there can be regions that HRT surfaces don’t reach The general expectation has been that EE can recover geometry wherever HRT surfaces reach, but never understood in detail

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Geometric Problem

Assumptions: Dimension of bulk geometry M is d ≥ 4, with finite boundary ∂M A portion R of M is foliated by a continuous (d − 2)-parameter family {Σ(λi)} of (planar) two-dimensional spacelike extremal surfaces anchored to ∂M

∂M Σ(λi)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Geometric Problem

Assumptions: Dimension of bulk geometry M is d ≥ 4, with finite boundary ∂M A portion R of M is foliated by a continuous (d − 2)-parameter family {Σ(λi)} of (planar) two-dimensional spacelike extremal surfaces anchored to ∂M

∂M Σ(λi)

Claim: the geometry in R is uniquely fixed by the metric and extrinsic curvature of ∂M, the curves ∂Σ(λi), and the variations of the areas of the Σ(λi)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Overview of Argument

Four steps, inspired by [Alexakis, Balehowsky, Nachman], using inverse boundary value problems (same sort of techniques used in e.g. medical imaging or geophysics)

1 Gauge fix: introduce a unique coordinate system {λi, xα} in the

region R, with the xα conformally flat coordinates on Σ(λi): ds2

Σ = e2φ[(dx1)2 + (dx2)2]

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Overview of Argument

Four steps, inspired by [Alexakis, Balehowsky, Nachman], using inverse boundary value problems (same sort of techniques used in e.g. medical imaging or geophysics)

1 Gauge fix: introduce a unique coordinate system {λi, xα} in the

region R, with the xα conformally flat coordinates on Σ(λi): ds2

Σ = e2φ[(dx1)2 + (dx2)2] 2 Showing that the gij ≡ gab(dλi)a(dλj)b are fixed reduces to an

elliptic inverse boundary value problem on each Σ(λi)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Overview of Argument

Four steps, inspired by [Alexakis, Balehowsky, Nachman], using inverse boundary value problems (same sort of techniques used in e.g. medical imaging or geophysics)

1 Gauge fix: introduce a unique coordinate system {λi, xα} in the

region R, with the xα conformally flat coordinates on Σ(λi): ds2

Σ = e2φ[(dx1)2 + (dx2)2] 2 Showing that the gij ≡ gab(dλi)a(dλj)b are fixed reduces to an

elliptic inverse boundary value problem on each Σ(λi)

3 By “tilting” each Σ(λi) to a nearby foliation, the

gαi ≡ gab(dxα)a(dλi)b are obtained by solving a system of (algebraic) linear equations (with known coefficients)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Overview of Argument

Four steps, inspired by [Alexakis, Balehowsky, Nachman], using inverse boundary value problems (same sort of techniques used in e.g. medical imaging or geophysics)

1 Gauge fix: introduce a unique coordinate system {λi, xα} in the

region R, with the xα conformally flat coordinates on Σ(λi): ds2

Σ = e2φ[(dx1)2 + (dx2)2] 2 Showing that the gij ≡ gab(dλi)a(dλj)b are fixed reduces to an

elliptic inverse boundary value problem on each Σ(λi)

3 By “tilting” each Σ(λi) to a nearby foliation, the

gαi ≡ gab(dxα)a(dλi)b are obtained by solving a system of (algebraic) linear equations (with known coefficients)

4 The requirement that the Σ(λi) all be extremal yields a

hyperbolic evolution equation for φ, which has a unique solution

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

The Jacobi (or Stability) Operator

Σ(s = 0) Σ(s)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

The Jacobi (or Stability) Operator

b b b

Σ(s = 0) Σ(s)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

The Jacobi (or Stability) Operator

b b b

Σ(s = 0) Σ(s) ηa ≡ (∂s)a

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

The Jacobi (or Stability) Operator

b b b

Σ(s = 0) Σ(s) ηa ≡ (∂s)a

If Σ(s) are all geodesics with tangent ta, deviation vector ηa obeys the equation of geodesic deviation 0 = tb∇b(tc∇cηa) + Rbcd

atbtdηc

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

The Jacobi (or Stability) Operator

b b b

Σ(s = 0) Σ(s) ηa ≡ (∂s)a

If Σ(s) are all extremal surfaces, deviation vector ηa obeys the Jacobi equation 0 = D2ηa +

• KacdKbcd + P acσdeRcdbe
• curvature terms ≡ Qab

ηb ≡ Jηa

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

The Jacobi (or Stability) Operator

Second variations of the area of an extremal surface Σ under deformations of ∂Σ give information about its Jacobi operator Extend Σ to an arbitrary two-parameter family Σ(s1, s2) of extremal surfaces, with Σ(0, 0) = Σ and deviation vectors ηa

1 = (∂s1)a, ηa 2 = (∂s2)a

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

The Jacobi (or Stability) Operator

Second variations of the area of an extremal surface Σ under deformations of ∂Σ give information about its Jacobi operator Extend Σ to an arbitrary two-parameter family Σ(s1, s2) of extremal surfaces, with Σ(0, 0) = Σ and deviation vectors ηa

1 = (∂s1)a, ηa 2 = (∂s2)a

The variation of the area A(s1, s2) is a boundary term: ∂2A ∂s1∂s2

• s1=0=s2

=

• ∂Σ

ηa

2DN(η1)a + (known boundary stuff)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

The Jacobi (or Stability) Operator

Second variations of the area of an extremal surface Σ under deformations of ∂Σ give information about its Jacobi operator Extend Σ to an arbitrary two-parameter family Σ(s1, s2) of extremal surfaces, with Σ(0, 0) = Σ and deviation vectors ηa

1 = (∂s1)a, ηa 2 = (∂s2)a

The variation of the area A(s1, s2) is a boundary term: ∂2A ∂s1∂s2

• s1=0=s2

=

• ∂Σ

ηa

2DN(η1)a + (known boundary stuff)

So knowing how the area varies as the shape of ∂Σ is varied yields the Dirichlet-to-Neumann map of J: Ψ : ηa|∂Σ → DNηa|∂Σ such that Jηa = 0

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Elliptic Inverse Boundary Value Problems fix gij

Inverse boundary value problem: if D†

1D1 + Q1 and D† 2D2 + Q2

acting on a vector bundle on a Riemann surface have the same Dirichlet-to-Neumann map, then D1, D2 and Q1, Q2 are the same (up to gauge) [Albin, Guillarmou, Tzou, Uhlmann]

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Elliptic Inverse Boundary Value Problems fix gij

Inverse boundary value problem: if D†

1D1 + Q1 and D† 2D2 + Q2

acting on a vector bundle on a Riemann surface have the same Dirichlet-to-Neumann map, then D1, D2 and Q1, Q2 are the same (up to gauge) [Albin, Guillarmou, Tzou, Uhlmann] So Jacobi operator J of each Σ(λi) is determined by boundary data up to choice of basis on the normal bundle

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Elliptic Inverse Boundary Value Problems fix gij

Inverse boundary value problem: if D†

1D1 + Q1 and D† 2D2 + Q2

acting on a vector bundle on a Riemann surface have the same Dirichlet-to-Neumann map, then D1, D2 and Q1, Q2 are the same (up to gauge) [Albin, Guillarmou, Tzou, Uhlmann] So Jacobi operator J of each Σ(λi) is determined by boundary data up to choice of basis on the normal bundle By construction, coordinate basis vector fields (∂λi)a are deviation vectors along a family of extremal surfaces, so J(∂λi)a = 0 Use this to fix the basis {(ni)a} on the normal bundle by requiring that (ni)a(∂λj)a = δij, which fixes (ni)a = (dλi)a

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Elliptic Inverse Boundary Value Problems fix gij

Inverse boundary value problem: if D†

1D1 + Q1 and D† 2D2 + Q2

acting on a vector bundle on a Riemann surface have the same Dirichlet-to-Neumann map, then D1, D2 and Q1, Q2 are the same (up to gauge) [Albin, Guillarmou, Tzou, Uhlmann] So Jacobi operator J of each Σ(λi) is determined by boundary data up to choice of basis on the normal bundle By construction, coordinate basis vector fields (∂λi)a are deviation vectors along a family of extremal surfaces, so J(∂λi)a = 0 Use this to fix the basis {(ni)a} on the normal bundle by requiring that (ni)a(∂λj)a = δij, which fixes (ni)a = (dλi)a Metric-compatibility of connection in this gauge requires Dagij = 0, which fixes gij

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Tilting Fixes gαi

Intuition: since the gαi know about “mixing” between directions normal and tangent to Σ(λi), we can mix them by “tilting” the folitation Σ(λi) to a family of foliations Σ(s; λi

s):

Σ(λi) ∂M

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Tilting Fixes gαi

Intuition: since the gαi know about “mixing” between directions normal and tangent to Σ(λi), we can mix them by “tilting” the folitation Σ(λi) to a family of foliations Σ(s; λi

s):

Σ(λi) Σ(s; λi

s)

∂M

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Tilting Fixes gαi

Intuition: since the gαi know about “mixing” between directions normal and tangent to Σ(λi), we can mix them by “tilting” the folitation Σ(λi) to a family of foliations Σ(s; λi

s):

Σ(λi) Σ(s; λi

s)

∂M

The {λi

s, xα s } give a new coordinate system (related to the

{λi, xα} by a diffeomorphism generated by ηa): λi

s(p) = λi(p)+sηi(p)+O(s2),

xα

s (p) = xα(p)+s ˙

xα(p)+O(s2)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Tilting Fixes gαi

Expanding gij

s ≡ gab(dλi s)a(dλj s)b to first order in s, get

dgij

s

ds

• s=0

− 2gk(i∂kηj) − ηk∂kgij

• known

= ˙ xα∂αgij + 2gα(i∂αηj)

• linear in unknowns ˙

xα, gαi

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Tilting Fixes gαi

Expanding gij

s ≡ gab(dλi s)a(dλj s)b to first order in s, get

dgij

s

ds

• s=0

− 2gk(i∂kηj) − ηk∂kgij

• known

= ˙ xα∂αgij + 2gα(i∂αηj)

• linear in unknowns ˙

xα, gαi

For d ≥ 4, there are enough linear equations to determine all the unknowns, and can recover gαi (d = 3 case studied by [Alexakis,

Balehowsky, Nachman] is much harder – need to compute the

deformation ˙ xα of the isothermal coordinates)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Hyperbolic PDE Fixes φ

Since gαβ = e2φδαβ, the extremality condition requires Ki = 0 ⇒ ∂αf α

i + 2f α i∂αφ − 2∂iφ = 0

(∗) with f α

i a known function of gij, gαi

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Hyperbolic PDE Fixes φ

Since gαβ = e2φδαβ, the extremality condition requires Ki = 0 ⇒ ∂αf α

i + 2f α i∂αφ − 2∂iφ = 0

(∗) with f α

i a known function of gij, gαi

Construct some periodic cycle in the λi, corresponding to a “tube” swept out by the Σ(λi)

∂M Σ(λi)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Hyperbolic PDE Fixes φ

Since gαβ = e2φδαβ, the extremality condition requires Ki = 0 ⇒ ∂αf α

i + 2f α i∂αφ − 2∂iφ = 0

(∗) with f α

i a known function of gij, gαi

Construct some periodic cycle in the λi, corresponding to a “tube” swept out by the Σ(λi)

∂M Σ(λi)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Hyperbolic PDE Fixes φ

Since gαβ = e2φδαβ, the extremality condition requires Ki = 0 ⇒ ∂αf α

i + 2f α i∂αφ − 2∂iφ = 0

(∗) with f α

i a known function of gij, gαi

Construct some periodic cycle in the λi, corresponding to a “tube” swept out by the Σ(λi)

∂M Σ(λi)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Hyperbolic PDE Fixes φ

Since gαβ = e2φδαβ, the extremality condition requires Ki = 0 ⇒ ∂αf α

i + 2f α i∂αφ − 2∂iφ = 0

(∗) with f α

i a known function of gij, gαi

Construct some periodic cycle in the λi, corresponding to a “tube” swept out by the Σ(λi) Evolve (∗) inwards from the boundary along this tube to fix φ uniquely on every Σ(λi) on it

∂M Σ(λi)

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Features

Applies to two-dimensional extremal surfaces in an ambient spacetime of any dimension (and signature), but relevance to bulk reconstruction is in d = 4, since then HRT surfaces are two-dimensional

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Features

Applies to two-dimensional extremal surfaces in an ambient spacetime of any dimension (and signature), but relevance to bulk reconstruction is in d = 4, since then HRT surfaces are two-dimensional Argument requires consistency of highly overdetermined systems

• f equations; this is expected on general grounds, as only the

right structure of entanglement entropy can possibly correspond to a geometric dual

Gives (highly implicit) necessary conditions for the existence of a dual geometry

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Features

Applies to two-dimensional extremal surfaces in an ambient spacetime of any dimension (and signature), but relevance to bulk reconstruction is in d = 4, since then HRT surfaces are two-dimensional Argument requires consistency of highly overdetermined systems

• f equations; this is expected on general grounds, as only the

right structure of entanglement entropy can possibly correspond to a geometric dual

Gives (highly implicit) necessary conditions for the existence of a dual geometry

Only requires knowledge of variations of entropy, not the actual entanglement entropy of any region

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Features

Can probe inside black holes!

H+ XR ∂M ∂M ∂M A H+ XR

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Features

Can probe inside black holes!

H+ XR ∂M ∂M ∂M A H+ XR

But still stays away from singularities...

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Reconstructive Formula

Our result is almost completely constructive; it gives a much more precise connection between boundary entanglement and bulk geometry than just HRT

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Reconstructive Formula

Our result is almost completely constructive; it gives a much more precise connection between boundary entanglement and bulk geometry than just HRT The only non-constructive step uses the uniqueness theorem of

[Albin, Guillarmou, Tzou, Uhlmann] to show that boundary data

uniquely fixes D and Q in the Jacobi operator D2 + Q; is there a constructive analog?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

A Reconstructive Formula

Our result is almost completely constructive; it gives a much more precise connection between boundary entanglement and bulk geometry than just HRT The only non-constructive step uses the uniqueness theorem of

[Albin, Guillarmou, Tzou, Uhlmann] to show that boundary data

uniquely fixes D and Q in the Jacobi operator D2 + Q; is there a constructive analog? Almost: a closely-related result gives an explicit method for recovering Q from the Dirichlet-to-Neumann map of the

• perator ∇2 + Q on some domain in R2, where ∇2 is the usual

(flat-space) Laplacian [Novikov, Santacesaria] Generalizing this to our case would give an explicit algorithm for recovering the metric from boundary entanglement

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Moving Further Up

The (semi)classical gravity we observe in our universe emerges from some more fundamental quantum theory - how? ⇓ (AdS/CFT) In AdS/CFT, how do the CFT degrees of freedom rearrange themselves to look like a gravitational theory? ⇓ (classical limit) When and how does (semi)classical gravity emerge from the boundary field theory? ⇓ (probe limit) How are operators on a fixed bulk geometry recovered?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Quantum Corrections to EE

Sub-leading effects in G (1/N 2 in CFT) introduce corrections: Engelhardt-Wall Formula Under perturbative quantum corrections, S[R] = Sgen[XR] ≡ Area[XR] 4G + Sout[XR], where XR is anchored to ∂R and extremizes Sgen (a “quantum extremal surface”), and Sout[XR] is the von Neumann entropy of any bulk quantum fields “outside” XR [Faulkner, Lewkowycz, Maldacena; Dong,

Lewkowycz]

R Sout[XR] XR

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Quantum Corrections to EE

Sub-leading effects in G (1/N 2 in CFT) introduce corrections: Engelhardt-Wall Formula Under perturbative quantum corrections, S[R] = Sgen[XR] ≡ Area[XR] 4G + Sout[XR], where XR is anchored to ∂R and extremizes Sgen (a “quantum extremal surface”), and Sout[XR] is the von Neumann entropy of any bulk quantum fields “outside” XR [Faulkner, Lewkowycz, Maldacena; Dong,

Lewkowycz]

R Sout[XR] XR (Note: XR can reach further into the bulk than XR, e.g. late-time horizons of evaporating black holes [Almheiri, Engelhardt, Marolf, Maxfield; Penington])

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Incorporating Quantum Effects

In order to really start probing quantum gravity effects, should be keeping track of these corrections! Can consider the same sort of setup, but with foliation by classical extremal surfaces XR replaced by quantum extremal surfaces XR

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Incorporating Quantum Effects

In order to really start probing quantum gravity effects, should be keeping track of these corrections! Can consider the same sort of setup, but with foliation by classical extremal surfaces XR replaced by quantum extremal surfaces XR Because Sout is not a local geometric functional, the Jacobi equation gets quantum-corrected to an integro-differential equation [Engelhardt, SF]: D2ηa + Qa

bηb + 4G

• Σ′ P ab

D2Sout DΣc(p′)DΣb ηc(p′) = 0, with DSout/DΣa a functional derivative

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Incorporating Quantum Effects

In order to really start probing quantum gravity effects, should be keeping track of these corrections! Can consider the same sort of setup, but with foliation by classical extremal surfaces XR replaced by quantum extremal surfaces XR Because Sout is not a local geometric functional, the Jacobi equation gets quantum-corrected to an integro-differential equation [Engelhardt, SF]: D2ηa +

• Σ′
• Qa

b(p, p′)ηb(p′) = 0,

with Qa

b(p, p′) a distributional potential

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Incorporating Quantum Effects

In order to really start probing quantum gravity effects, should be keeping track of these corrections! Can consider the same sort of setup, but with foliation by classical extremal surfaces XR replaced by quantum extremal surfaces XR Because Sout is not a local geometric functional, the Jacobi equation gets quantum-corrected to an integro-differential equation [Engelhardt, SF]: D2ηa +

• Σ′
• Qa

b(p, p′)ηb(p′) = 0,

with Qa

b(p, p′) a distributional potential

Are D and Qa

b determined by boundary data just as they are in

the classical case? If so, can generalize argument

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Higher Curvature Corrections

Turning on α′ corrections changes the bulk gravitational dynamics to include higher-curvature corrections

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Higher Curvature Corrections

Turning on α′ corrections changes the bulk gravitational dynamics to include higher-curvature corrections HRT formula changes: area functional becomes another geometric functional [Dong; Camps] Perturbations give rise to a generalized Jacobi operator J that depends on the perturbed area functional

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Higher Curvature Corrections

Turning on α′ corrections changes the bulk gravitational dynamics to include higher-curvature corrections HRT formula changes: area functional becomes another geometric functional [Dong; Camps] Perturbations give rise to a generalized Jacobi operator J that depends on the perturbed area functional If J be recovered from boundary data, can likewise generalize the argument to recover the bulk even when it includes these higher-curvature corrections

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Summary

Argued that for a 4d classical bulk, the bulk geometry near an HRT surface is fixed by entanglement entropy of the boundary, nicely connecting bulk reconstruction in AdS/CFT and inverse boundary value problems

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Summary

Argued that for a 4d classical bulk, the bulk geometry near an HRT surface is fixed by entanglement entropy of the boundary, nicely connecting bulk reconstruction in AdS/CFT and inverse boundary value problems To actually start probing quantum effects, need to extend these results to (i) an explicit reconstruction that (ii) includes perturbative quantum corrections to the HRT formula; the framework for both exists, and this work is ongoing

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Summary

Argued that for a 4d classical bulk, the bulk geometry near an HRT surface is fixed by entanglement entropy of the boundary, nicely connecting bulk reconstruction in AdS/CFT and inverse boundary value problems To actually start probing quantum effects, need to extend these results to (i) an explicit reconstruction that (ii) includes perturbative quantum corrections to the HRT formula; the framework for both exists, and this work is ongoing Even geometry where HRT surfaces don’t reach should be recoverable somehow; from what? Quantum extremal surfaces? Other measures of entanglement?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Context Holographic EE Geometric Bulk Reconstruction Extensions

Summary

Argued that for a 4d classical bulk, the bulk geometry near an HRT surface is fixed by entanglement entropy of the boundary, nicely connecting bulk reconstruction in AdS/CFT and inverse boundary value problems To actually start probing quantum effects, need to extend these results to (i) an explicit reconstruction that (ii) includes perturbative quantum corrections to the HRT formula; the framework for both exists, and this work is ongoing Even geometry where HRT surfaces don’t reach should be recoverable somehow; from what? Quantum extremal surfaces? Other measures of entanglement? More generalizations: (d − 2)-dimensional surfaces in higher dimension d; higher-curvature corrections; how generic is the assumption of a foliation?

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Fixing Conformally Flat Coordinates

y1 y2 gαβ gαβ = δαβ Introduce arbitrary coordinate system {yα} on Σ ⊂ R2

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Fixing Conformally Flat Coordinates

y1 y2 gαβ gαβ = δαβ Introduce arbitrary coordinate system {yα} on Σ ⊂ R2 Extend gαβ to all of R2 so that gαβ = δαβ away from Σ, and gαβ is known everywhere outside Σ There exists a unique set of isothermal coordinates {xα} on R2 such that xα(y) → yα at large yα [Ahlfors]

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Fixing Conformally Flat Coordinates

y1 y2 gαβ gαβ = δαβ Introduce arbitrary coordinate system {yα} on Σ ⊂ R2 Extend gαβ to all of R2 so that gαβ = δαβ away from Σ, and gαβ is known everywhere outside Σ There exists a unique set of isothermal coordinates {xα} on R2 such that xα(y) → yα at large yα [Ahlfors] For any g1, g2 on Σ with the same Ψ, the corresponding coordinates xα

1 (y), xα 2 (y)

agree outside Σ and on ∂Σ

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement

Fixing Conformally Flat Coordinates

y1 y2 gαβ gαβ = δαβ Introduce arbitrary coordinate system {yα} on Σ ⊂ R2 Extend gαβ to all of R2 so that gαβ = δαβ away from Σ, and gαβ is known everywhere outside Σ There exists a unique set of isothermal coordinates {xα} on R2 such that xα(y) → yα at large yα [Ahlfors] For any g1, g2 on Σ with the same Ψ, the corresponding coordinates xα

1 (y), xα 2 (y)

agree outside Σ and on ∂Σ So for any two metrics g1, g2 on Σ with the same boundary data, there exists a set of coordinates {xα} on Σ in which both are conformally flat: ds2

Σ = e2φ

(dx1)2 + (dx2)2

Sebastian Fischetti McGill University Recovering a Holographic Geometry from Entanglement